Questions tagged [central-limit-theorem]

This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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A die is thrown until the first time the total sum of the face values of the die is 700 or greater. What is the probability for $n$ tosses?

Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required We're supposed to be using ...
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Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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Show $Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$

Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly ...
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If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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Domain of Central Limit Theorem

The central limit theorem says that if you take infinite number of samples ( > 30) from a population, compute their mean values, and collect them, you will reach normal distribution. Is this valid for ...
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An example where Lyapounov Condition is much easier than the Lindeberg?

Currently studying the Lindeberg-Levy-Feller and Lyapounov's Central Limit Theorems, and was wondering - for sake of justification - are there any examples where verifying Lyapounov's condition is ...
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A central limit theorem for dependent random variable.

Suppose that $u_{j}$ is a sequence of iid standard Gaussian random variable, i.e. $$u_j\stackrel{d}{=}\text{N}(0,1).$$ Call $\mu_r=\mathbb{E}[|u_j|^r]$. I need to find the asymptotic distribution ...
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CLT for decaying random variables

Suppose $X_1,X_2,...$ are bounded random variables with compact support, and $\frac{X_1+...+X_n}{\sqrt{n}}\overset{d}{\longrightarrow}N(0,1)$. Is there neccessarily a central limit theorem for the ...
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Central limit theorem … need help

So far I have used Chebychev's inequality to calculate n = 157, and my initial thinking for applying the central limit theorem is -1.2815 = (5-0n)/(n* sqrt(391/n)) because P(Z > -1.2815) = 90%, ...
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Proof that if you take enough steps of equal magnitude on a plane, you'll always end up at the starting point

Is the claim in the title true? If yes, is the following sufficient to justify it intuitively? Assume for simplicity that the steps are performed with magnitude 1 in a Cartesian plane. To construct a ...
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using Central Limit Theorem to approximate a probability with large 'n'

For $i>1$ , let $X_i$ ~ $G_{1/2}$ be distributed Geometrically with parameter $1/2$. $$Y_n= \frac{1}{\sqrt n} \sum_{r=1}^n X_r-2$$ Approximate $P(-1\le Y_n\le 2)$ with large n.($Y_n$ is not ...
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